Eurozone economy commissioner Olli Rehn is right to insist that this is no time for any complacency, Photograph: Thierry Roge/EPA

Many of Europe's leaders will have hit the sun loungers for their customary August break by the time last week's GDP data was published, but news that the eurozone had emerged from recession must have perked up the holiday mood no end.

A strong performance by Germany helped to deliver 0.3% growth across the 17-member single currency bloc in the second quarter of this year, bringing an 18-month recession to a close.

This improvement in the economic weather may partly have resulted from the more lenient approach the "troika" of the International Monetary Fund, European Central Bank and European Commission has adopted in recent months. Several countries, including France, were given extra time to meet fiscal targets, letting their governments temper the savage austerity plans that threatened to prolong the downturn.

But while the darkest hour may have passed, there are at least three reasons to worry that the crisis is far from over.

First, despite the encouraging data, the economic fundamentals in most countries remain grim. While domestic consumption finally appears to be on the rise in Germany, which should help to boost exports from other member countries, many markets further afield remain fragile.

The trade balances of some of the crisis-hit states have started to improve, but often more because of collapsing imports than a healthy surge in exports. "Where are the growth drivers going to come from?" asked Neil Mellor of BNY Mellon. "It's still export-dependent and that means the rest of the eurozone is feeding on the scraps from Germany's table; and Germany is dependent on China, which is going south."

The peripheral countries continue to lag far behind the wealthier core, with several still in recession and many saddled with eye-watering rates of unemployment: Greece 26.9%, Spain 26.3%, Ireland 13.5%.

Peter Shaffrik of RBC Capital Markets argued last week that one of the key lessons from five years of crisis has been that "smaller economies in Europe – even closed ones like Greece – can dramatically affect prospects for the currency union as a whole." Second, and relatedly, the debt dynamics in several countries look unsustainable, even if growth is restored. Greece has a "financing gap" of up to €11bn (£9.4bn), according to the IMF, which will have to be filled through fresh austerity measures or another round of debt restructuring. Portugal looks unlikely to be able to graduate from its IMF programme in 2014 and return to the bond markets as its creditors hoped; and Italy and Spain are still not out of the danger zone.

If bond yields worldwide continue to drift up in response to fears that the US Federal Reserve is about to choke off the supply of cheap money to investors by phasing out its quantitative easing programme, it could drive up funding costs for countries already under pressure. The IMF highlighted this risk in its recent annual health check of the eurozone.

Third, progress on many of the most pressing political problems, including completing a banking union, are unlikely to be resolved until after the German elections in September.

Linking together the region's banks under a single regulator – the ECB – and establishing a common rescue fund to bail them out if they fail could act as a circuit-breaker by severing the lethal connection between countries' public finances and the health of their banks.

But there are deep disagreements between member countries about how the new structures will work. Germany responded to the latest proposals from internal market commissioner Michel Barnier – which would involve a new Brussels body deciding when a struggling bank should be wound up – by arguing that creating such an institution would be a breach of the EU treaty. Wrangling over these issues is likely to continue for many months, with the constant potential to reignite market panic.

Finland's Olli Rehn, the economy commissioner, insisted after the data was published last week that there was "no time for any complacency whatsoever". He was right.